Peter Jipsen

Algebra, Logic and Foundations of Mathematics, Theory of Computation

Ph. D.
16.57

Publications

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    Peter Jipsen · M. Andrew Moshier
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    ABSTRACT: We introduce concurrent Kleene algebra with tests (CKAT) as a combination of Kleene algebra with tests (KAT) of Kozen and Smith with concurrent Kleene algebras (CKA), introduced by Hoare, Möller, Struth and Wehrman. CKAT provides a relatively simple algebraic model for reasoning about semantics of concurrent programs. We generalize guarded strings to \emph{guarded series-parallel strings}, or gsp-strings, to give a concrete language model for CKAT. Combining nondeterministic guarded automata of Kozen with branching automata of Lodaya and Weil one obtains a model for processing gsp-strings in parallel. To ensure that the model satisfies the weak exchange law $(x||y)(z||w)\leq(xz)||(yw)$ of CKA, we make use of the subsumption order of Gischer on the gsp-strings. We also define \emph{deterministic} branching automata and investigate their relation to (nondeterministic) branching automata. To express basic concurrent algorithms, we define concurrent deterministic flowchart schemas and relate them to branching automata and to concurrent Kleene algebras with tests.
    Full-text · Article · Dec 2015
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    M. ANDREW MOSHIER · PETER JIPSEN
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    ABSTRACT: Lattices have many applications in mathematics and logic, in which they occur together with additional operations. For example, in applications of Hilbert spaces, one is often con- cerned with the lattice of closed subspaces of a fixed space. This lattice is not distributive, but there is an operation taking a given subspace to its orthogonal subspace. More gen- erally, ortholattices are lattices with a unary operation ( )y that is involutive (a = ayy), sends finite joins to meets and for which a and ay are complements. Bounded modal lat- tices (L;_;^; 0; 1; ;2) are models of (not necessarily distributive) modal logic, where and 2 are unary operations that preserve finite join and finite meet, respectively, and represent possible and necessary. Bounded lattice-ordered monoids are bounded lattices with an associative binary operation and an identity element 1. In these examples it is postulated that the additional operations "preserve structure" in various different senses. Orthocomplementation sends finite joins to meets (and finite meets to joins). The modal operators preserve finite joins and finite meets, respectively. Similarly, the monoid oper- ation distribute over finite joins. Bounded residuated lattices are bounded lattice-ordered monoids with two further operationsn, = that interact with via the universally quantified residuation law:
    Full-text · Article · May 2014 · Algebra Universalis
  • Peter Jipsen
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    ABSTRACT: Concurrent Kleene algebras were introduced by Hoare, Möller, Struth and Wehrman in [HMSW09, HMSW09a, HMSW11] as idempotent bisemirings that satisfy a concurrency inequation and have a Kleene-star for both sequential and concurrent composition. Kleene algebra with tests (KAT) were defined earlier by Kozen and Smith [KS97]. Concurrent Kleene algebras with tests (CKAT) combine these concepts and give a relatively simple algebraic model for reasoning about operational semantics of concurrent programs. We generalize guarded strings to guarded series-parallel strings, or gsp-strings, to provide a concrete language model for CKAT. Combining nondeterministic guarded automata [Koz03] with branching automata of Lodaya and Weil [LW00] one obtains a model for processing gsp-strings in parallel, and hence an operational interpretation for CKAT. For gsp-strings that are simply guarded strings, the model works like an ordinary nondeterministic guarded automaton. If the test algebra is assumed to be {0,1} the language model reduces to the regular sets of bounded-width sp-strings of Lodaya and Weil. Since the concurrent composition operator distributes over join, it can also be added to relation algebras with transitive closure to obtain the variety CRAT. We provide semantics for these algebras in the form of coalgebraic arrow frames expanded with concurrency.
    No preview · Article · Apr 2014

  • No preview · Conference Paper · Jan 2014
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    Peter Jipsen · Nathan Lawless
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    ABSTRACT: Modular lattices, introduced by R. Dedekind, are an important subvariety of lattices that includes all distributive lattices. Heitzig and Reinhold developed an algorithm to enumerate, up to isomorphism, all finite lattices up to size 18. Here we adapt and improve this algorithm to construct and count modular lattices up to size 23, semimodular lattices up to size 22, and lattices of size 19. We also show that $2^{n-3}$ is a lower bound for the number of nonisomorphic modular lattices of size $n$.
    Full-text · Article · Sep 2013 · Algebra Universalis
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    Nikolaos Galatos · Peter Jipsen
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    ABSTRACT: Residuated frames provide relational semantics for substructural logics and are a natural generalization of Kripke frames in intuitionistic and modal logic, and of phase spaces in linear logic. We explore the connection between Gentzen systems and residuated frames and illustrate how,frames provide a uniform treatment for semantic proofs of cut-elimination, the finite model property and the finite embeddability property. We use our results to prove the decidability of the equational and/or universal theory of several vari- eties of residuated lattice-ordered groupoids, including the variety of involutive FL-algebras. Substructural logics and their algebraic formulation as varieties of residuated
    Full-text · Article · Mar 2013 · Transactions of the American Mathematical Society
  • Nikolaos Galatos · Peter Jipsen
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    ABSTRACT: This paper studies generalizations of relation algebras to residuated lattices with a unary De Morgan operation. Several new examples of such algebras are presented, and it is shown that many basic results on relation algebras hold in this wider setting. The variety qRA of quasi relation algebras is defined and shown to be a conservative expansion of involutive FL-algebras. Our main result is that equations in qRA and several of its subvarieties can be decided by a Gentzen system, and that these varieties are generated by their finite members.
    No preview · Article · Feb 2013 · Algebra Universalis
  • Nikolaos Galatos · Peter Jipsen · Hiroakira Ono

    No preview · Article · Dec 2012 · Studia Logica
  • Nikolaos Galatos · Peter Jipsen
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    ABSTRACT: It is proved that any lattice-ordered pregroup that satisfies an identity of the form x ll···l = x rr···r (for the same number of l,r -operations on each side) has a lattice reduct that is distributive. It follows that every such ℓ-pregroup is embedded in an ℓ-pregroup of residuated and dually residuated maps on a chain.
    No preview · Article · Oct 2012 · Algebra Universalis
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    Peter Jipsen
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    ABSTRACT: A categorical equivalence between algebraic contexts with relational morphisms and join-semilattices with homomorphisms is presented and extended to idempotent semirings and domain semirings. These contexts are the Kripke structures for idempotent semirings and allow more efficient computations on finite models because they can be logarithmically smaller than the original semiring. Some examples and constructions such as matrix semirings are also considered.
    Full-text · Conference Paper · Sep 2012
  • Article: Preface
    Nikolaos Galatos · Peter Jipsen · Hiroakira Ono

    No preview · Article · Jan 2012 · Studia Logica
  • Peter Jipsen · Chris Brink · Gunther Schmidt
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    ABSTRACT: This chapter serves the rest of the book: all later chapters presuppose it. It introduces the calculus of binary relations, and relates it to basic concepts and results from lattice theory, universal algebra, category theory and logic. It also fixes the notation and terminology to be used in the rest of the book. Our aim here is to write in a way accessible to readers who desire a gentle introduction to the subject of relational methods. Other readers may prefer to go on to further chapters, only referring back to Chapt. 1 as needed.
    No preview · Chapter · Jul 2011
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    Peter Jipsen · Henry Rose
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    ABSTRACT: An interesting problem in universal algebra is the connection between the internal structure of an algebra and the identities which it satisfies. The study of varieties of algebras provides some insight into this problem. Here we are concerned mainly with lattice varieties, about which a wealth of information has been obtained in the last twenty years. We begin with some preliminary results from universal algebra and lattice theory. The second chapter presents some properties of the lattice of all lattice subvarieties. Here we also discuss the important notion of a splitting pair of varieties and give several characterizations of the associated splitting lattice. The more detailed study of lattice varieties splits naturally into the study of modular lattice varieties and nonmodular lattice varieties, dealt with in the third and fourth chapter respectively. Among the results discussed there are Freese's theorem that the variety of all modular lattices is not generated by its finite members, and several results concerning the question which varieties cover a given variety. The fifth chapter contains a proof of Baker's finite basis theorem and some results about the join of finitely based lattice varieties. Included in the final chapter is a characterization of the amalgamation classes of certain congruence distributive varieties and the result that
    Full-text · Article · Sep 2010
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    P. Jipsen · F. Montagna
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    ABSTRACT: The poset product construction is used to derive embedding theorems for several classes of generalized basic logic algebras (GBL-algebras). In particular it is shown that every n-potent GBL-algebra is embedded in a poset product of finite n-potent MV-chains, and every normal GBL-algebra is embedded in a poset product of totally ordered GMV-algebras. Representable normal GBL-algebras have poset product embeddings where the poset is a root system. We also give a Conrad–Harvey–Holland-style embedding theorem for commutative GBL-algebras, where the poset factors are the real numbers extended with −∞. Finally, an explicit construction of a generic commutative GBL-algebra is given, and it is shown that every normal GBL-algebra embeds in the conucleus image of a GMV-algebra.
    Full-text · Article · Sep 2010 · Journal of Pure and Applied Algebra
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    Jules Desharnais · Peter Jipsen · Georg Struth
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    ABSTRACT: We axiomatise and study operations for relational domain and antidomain on semigroups and monoids. We relate this approach with previous axiomatisations for semirings, partial transformation semi- groups and dynamic predicate logic.
    Full-text · Conference Paper · Nov 2009
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    Peter Jipsen
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    ABSTRACT: It is shown that the Boolean center of complemented elements in a bounded in- tegral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FLw-algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded n-potent GBL-algebras are represented as Esakia products of simple n-potent MV-algebras.
    Full-text · Article · Nov 2009 · Annals of Pure and Applied Logic
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    P. Jipsen · F. Montagna
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    ABSTRACT: Generalized basic logic algebras (GBL-algebras for short) have been introduced in [JT02] as a generalization of Hájek’s BL-algebras, and constitute a bridge between algebraic logic and ℓ-groups. In this paper we investigate normal GBL-algebras, that is, integral GBL-algebras in which every filter is normal. For these structures we prove an analogue of Blok and Ferreirim’s [BF00] ordinal sum decomposition theorem. This result allows us to derive many interesting consequences, such as the decidability of the universal theory of commutative GBL-algebras, the fact that n-potent GBL-algebras are commutative, and a representation theorem for finite GBL-algebras as poset sums of GMV-algebras, a result which generalizes Di Nola and Lettieri’s [DL03] representation of finite BL-algebras.
    Full-text · Article · May 2009 · Algebra Universalis
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    Peter Jipsen

    Full-text · Article · Jan 2009
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    M. ANDREW MOSHIER · PETER JIPSEN
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    ABSTRACT: The two main objectives of this paper are (a) to prove purely topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. In previously known dualities for semilattices and bounded lattices, the dual spaces are compact 0-dimensional spaces with additional algebraic structure. For example, semilattices are dual to 0-dimensional compact semilattices. Here we establish dual categories in which the spaces are characterized purely in topological terms, with no additional algebraic structure. Thus the results can be seen as generalizing Stone's duality for distributive lattices rather than Priestley's. The paper is the first of two parts. The main objective of the sequel is to establish a characterization of lattice expansions, i.e., lattices with additional operations, in the topological setting built in this paper.
    Full-text · Article · Jan 2009 · Algebra Universalis
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    Nikolaos Galatos · Peter Jipsen
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    ABSTRACT: Petr Hajek identified the logic BL, that was later shown to be the logic of continuous t-norms on the unit interval, and defined the corresponding algebraic models, BL-algebras, in the context of residuated lattices. The defining characteristics of BL-algebras are representability and divisibility. In this short note we survey re- cent developments in the study of divisible residuated lattices and attribute the inspiration for this investigation to Petr Hajek.
    Full-text · Chapter · Jan 2009

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